LMI-Based Lower Bound Analysis of the Best Achievable <i>H</i><sub>∞</sub> Performance for SISO Systems

Yoshio Ebihara, Keisuke Matsuo, Tomomichi Hagiwara

Research output: Contribution to journalArticlepeer-review


In this paper, we study <i>H</i><sub>∞</sub> performance limitation analysis for continuous-time SISO systems using LMIs. By starting from an LMI that characterizes a necessary and sufficient condition for the existence of desired controllers achieving a prescribed <i>H</i><sub>∞</sub> performance level, we represent lower bounds of the best <i>H</i><sub>∞</sub> performance achievable by any LTI controller in terms of the unstable zeros and the unstable poles of a given plant. The transfer functions to be investigated include the sensitivity function (1+<i>PK</i>)<sup>-1</sup>, the complementary sensitivity function (1+<i>PK</i>)<sup>-1</sup><i>PK</i>, and (1+<i>PK</i>)<sup>-1</sup><i>P</i>, the first and the second of which are well investigated in the literature. As a main result, we derive lower bounds of the best achievable <i>H</i><sub>∞</sub> performance with respect to (1+<i>PK</i>)<sup>-1</sup><i>P</i> assuming that the plant has unstable zeros. More precisely, we characterize a lower bound in closed-form by means of the first non-zero coefficient of the Taylor expansion of the plant <i>P</i>(<i>s</i>) around its unstable zero.
Original languageEnglish
Pages (from-to)165-172
Number of pages8
JournalSICE Journal of Control, Measurement, and System Integration
Issue number4
Publication statusPublished - 2016


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